[[Topological group]]
# Haar measure

Given a [[Locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] $G$,
there exist [[Trivial measure|nontrivial]], [[Regular measure|regular]], [[Locally finite measure|locally finite]], left- and right-invariant [[Borel set|Borel measures]] on $G$,
unique up to multiplication by a positive constant,
called the **left Haar measure** $\mu_{L}$ and **right Haar measure** $\mu_{R}$ of $G$ respectively. #m/thm/group
Invariance means given [[Borel set]] $U \sube G$ and a group element $g \in G$

- $\mu_{L}(U) = \mu_{L}(gU)$
- $\mu_{R}(U) = \mu_{R}(Ug)$


> [!missing]- Proof
> #missing/proof Too advanced for now

The main use of the Haar measure is analogous to the [[Reärrangement lemma]] in finite contexts.

## Further terminology

- A [[Unimodular group]] is a group for which $\mu_{L}=\mu_{R} = \mu$.

## Explicit constructions and examples

- [[Haar measure of a discrete group]] is counting measure
- [[Haar measure of a compact Lie group]]

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